Electron. Colloquium Comput. Complex.最新文献

{"title":"New Lower Bounds against Homogeneous Non-Commutative Circuits","authors":"Prerona Chatterjee, Pavel Hrubevs","doi":"10.48550/arXiv.2301.01676","DOIUrl":"https://doi.org/10.48550/arXiv.2301.01676","url":null,"abstract":"We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $Omega(d/log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $Omega(nd)$, if $dleq n$, or $Omega(nd frac{log n}{log d})$, if $dgeq n$. Under the same assumptions, we also give a quadratic lower bound for the ordered version of the central symmetric polynomial.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"58 1","pages":"13:1-13:10"},"PeriodicalIF":0.0,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76020974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Near-Optimal Set-Multilinear Formula Lower Bounds","authors":"D. Kush, Shubhangi Saraf","doi":"10.4230/LIPIcs.CCC.2023.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.15","url":null,"abstract":"The seminal work of Raz (J. ACM 2013) as well as the recent breakthrough results by Limaye, Srinivasan, and Tavenas (FOCS 2021, STOC 2022) have demonstrated a potential avenue for obtaining lower bounds for general algebraic formulas, via strong enough lower bounds for set-multilinear formulas. In this paper, we make progress along this direction by proving near-optimal lower bounds against low-depth as well as unbounded-depth set-multilinear formulas. More precisely, we show that over any field of characteristic zero, there is a polynomial f computed by a polynomial-sized set-multilinear branching program (i.e., f is in set-multilinear VBP ) defined over Θ( n 2 ) variables and of degree Θ( n ), such that any product-depth ∆ set-multilinear formula computing f has size at least n Ω( n 1 / ∆ / ∆) . Moreover, we show that any unbounded-depth set-multilinear formula computing f has size at least n Ω(log n ) . If such strong lower bounds are proven for the iterated matrix multiplication (IMM) polynomial or rather, any polynomial that is computed by an ordered set-multilinear branching program (i.e., a further restriction of set-multilinear VBP), then this would have dramatic consequences as it would imply super-polynomial lower bounds for general algebraic formulas (Raz, J. ACM 2013; Tavenas, Limaye, and Srinivasan, STOC 2022). Prior to our work, either only weaker lower bounds were known for the IMM polynomial (Tavenas, Limaye, and Srinivasan, STOC 2022), or similar strong lower bounds were known but for a hard polynomial not known to be even in set-multilinear VP (Kush and Saraf, CCC 2022; Raz, J. ACM","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77500590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Efficient Interactive Proofs for Non-Deterministic Bounded Space","authors":"Joshua Cook, R. Rothblum","doi":"10.4230/LIPIcs.APPROX/RANDOM.2023.47","DOIUrl":"https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.47","url":null,"abstract":"The celebrated IP = PSPACE Theorem gives an efficient interactive proof for any bounded-space algorithm. In this work we study interactive proofs for non-deterministic bounded space computations. While Savitch’s Theorem shows that nondeterministic bounded-space algorithms can be simulated by deterministic bounded-space algorithms, this simulation has a quadratic overhead. We give interactive protocols for nondeterministic algorithms directly to get faster verifiers. More specifically, for any non-deterministic space S algorithm, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S 2 ). This improves on the best previous bound of ˜ O ( n + S 3 ) and matches the result for deterministic space bounded algorithms, up to polylog( S ) factors. We further generalize to alternating bounded space algorithms. For any language L decided by a time T , space S algorithm that uses d alternations, we construct an interactive proof in which the verifier runs in time ˜ O ( n + S log( T ) + Sd ) and the prover runs in time 2 O ( S ) . For d = O (log( T )), this matches the best known interactive proofs for deterministic algorithms, up to polylog( S ) factors, and improves on the previous best verifier time for nondeterministic algorithms by a factor of log( T ). We also improve the best prior verifier time for unbounded alternations by a factor of S . Using known connections of bounded alternation algorithms to bounded depth circuits, we also obtain faster verifiers for bounded depth circuits with unbounded fan-in.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88674689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"QCDCL vs QBF Resolution: Further Insights","authors":"Benjamin Böhm, Olaf Beyersdorff","doi":"10.4230/LIPIcs.SAT.2023.4","DOIUrl":"https://doi.org/10.4230/LIPIcs.SAT.2023.4","url":null,"abstract":"We continue the investigation on the relations of QCDCL and QBF resolution systems. In particular, we introduce QCDCL versions that tightly characterise QU-Resolution and (a slight variant of) long-distance Q-Resolution. We show that most QCDCL variants – parameterised by different policies for decisions, unit propagations and reductions – lead to incomparable systems for almost all choices of these policies.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"17 1","pages":"4:1-4:17"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84317267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Derandomization with Minimal Memory Footprint","authors":"Dean Doron, R. Tell","doi":"10.4230/LIPIcs.CCC.2023.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.11","url":null,"abstract":"Existing proofs that deduce BPL = L from circuit lower bounds convert randomized algorithms into deterministic algorithms with large constant overhead in space. We study space-bounded derandomization with minimal footprint, and ask what is the minimal possible space overhead for derandomization. We show that BPSPACE [ S ] ⊆ DSPACE [ c · S ] for c ≈ 2, assuming space-efficient cryptographic PRGs, and, either: (1) lower bounds against bounded-space algorithms with advice, or: (2) lower bounds against certain uniform compression algorithms. Under additional assumptions regarding the power of catalytic computation, in a new setting of parameters that was not studied before, we are even able to get c ≈ 1. Our results are constructive: Given a candidate hard function (and a candidate cryptographic PRG) we show how to transform the randomized algorithm into an efficient deterministic one. This follows from new PRGs and targeted PRGs for space-bounded algorithms, which we combine with novel space-efficient evaluation methods. A central ingredient in all our constructions is hardness amplification reductions in logspace-uniform TC 0 , that were not known before. 2012 ACM Subject Classification Theory of computation → Complexity theory and logic; Theory of computation → Pseudorandomness and derandomization; Theory of computation → Error-correcting codes","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"105 1","pages":"11:1-11:15"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73736197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Colourful TFNP and Propositional Proofs","authors":"B. Davis, Robert Robere","doi":"10.4230/LIPIcs.CCC.2023.36","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.36","url":null,"abstract":"Recent work has shown that many of the standard TFNP classes – such as PLS , PPADS , PPAD , SOPL , and EOPL – have corresponding proof systems in propositional proof complexity, in the sense that a total search problem is in the class if and only if the totality of the problem can be efficiently proved by the corresponding proof system. We build on this line of work by studying coloured variants of these TFNP classes: C - PLS , C - PPADS , C - PPAD , C - SOPL , and C - EOPL . While C - PLS has been studied in the literature before, the coloured variants of the other classes are introduced here for the first time. We give a family of results showing that these coloured TFNP classes are natural objects of study, and that the correspondence between TFNP and natural propositional proof systems is not an exceptional phenomenon isolated to weak TFNP classes. Namely, we show that: Each of the classes C - PLS , C - PPADS , and C - SOPL have corresponding proof systems characterizing them. Specifically, the proof systems for these classes are obtained by adding depth to the formulas in the corresponding proof system for the uncoloured class. For instance, while it was previously known that PLS is characterized by bounded-width Resolution (i.e. depth 0.5 Frege), we prove that C - PLS is characterized by depth-1.5 Frege (Res( polylog ( n ))). The classes C - PPAD and C - EOPL coincide exactly with the uncoloured classes PPADS and SOPL , respectively. Thus, both of these classes also have corresponding proof systems: unary Sherali-Adams and Reversible Resolution, respectively. Finally, we prove a coloured intersection theorem for the coloured sink classes, showing C - PLS ∩ C - PPADS = C - SOPL , generalizing the intersection theorem PLS ∩ PPADS = SOPL . However, while it is known in the uncoloured world that PLS ∩ PPAD = EOPL = CLS , we prove that this equality fails in the coloured world in the black-box setting. More precisely, we show that there is an oracle O such that C - PLS O ∩ C - PPAD O ⊋ C - EOPL O . To prove our results, we introduce an abstract multivalued proof system – the Blockwise Calculus – which may be of independent interest.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"46 1","pages":"36:1-36:21"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86685083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Instance-Wise Hardness versus Randomness Tradeoffs for Arthur-Merlin Protocols","authors":"Nicollas M. Sdroievski, D. Melkebeek","doi":"10.4230/LIPIcs.CCC.2023.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.17","url":null,"abstract":"A fundamental question in computational complexity asks whether probabilistic polynomial-time algorithms can be simulated deterministically with a small overhead in time (the BPP vs. P problem). A corresponding question in the realm of interactive proofs asks whether Arthur-Merlin protocols can be simulated nondeterministically with a small overhead in time (the AM vs. NP problem). Both questions are intricately tied to lower bounds. Prominently, in both settings blackbox derandomization, i.e., derandomization through pseudo-random generators, has been shown equivalent to lower bounds for decision problems against circuits. Recently, Chen and Tell (FOCS’21) established near-equivalences in the BPP setting between whitebox derandomization and lower bounds for multi-bit functions against algorithms on almost-all inputs. The key ingredient is a technique to translate hardness into targeted hitting sets in an instance-wise fashion based on a layered arithmetization of the evaluation of a uniform circuit computing the hard function f on the given instance. In this paper we develop a corresponding technique for Arthur-Merlin protocols and establish similar near-equivalences in the AM setting. As an example of our results in the hardness to derandomization direction, consider a length-preserving function f computable by a nondeterministic algorithm that runs in time n a . We show that if every Arthur-Merlin protocol that runs in time n c for c = O (log 2 a ) can only compute f correctly on finitely many inputs, then AM is in NP . Our main technical contribution is the construction of suitable targeted hitting-set generators based on probabilistically checkable proofs for nondeterministic computations. As a byproduct of our constructions, we obtain the first result indicating that whitebox derandomization of AM may be equivalent to the existence of targeted hitting-set generators for AM , an issue raised by Goldreich (LNCS, 2011). Byproducts in the average-case setting include the first uniform hardness vs. randomness tradeoffs for AM , as well as an unconditional mild derandomization result for AM .","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"2010 1","pages":"17:1-17:36"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82546818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reducing Tarski to Unique Tarski (in the Black-box Model)","authors":"Xi Chen, Yuhao Li, M. Yannakakis","doi":"10.4230/LIPIcs.CCC.2023.21","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.21","url":null,"abstract":"We study the problem of finding a Tarski fixed point over the k -dimensional grid [ n ] k . We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88239665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bounded Relativization","authors":"Shuichi Hirahara, Zhenjian Lu, Hanlin Ren","doi":"10.4230/LIPIcs.CCC.2023.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.6","url":null,"abstract":"Relativization is one of the most fundamental concepts in complexity theory, which explains the difficulty of resolving major open problems. In this paper, we propose a weaker notion of relativization called bounded relativization . For a complexity class C , we say that a statement is C -relativizing if the statement holds relative to every oracle O ∈ C . It is easy to see that every result that relativizes also C -relativizes for every complexity class C . On the other hand, we observe that many non-relativizing results, such as IP = PSPACE , are in fact PSPACE -relativizing. First, we use the idea of bounded relativization to obtain new lower bound results, including the following nearly maximum circuit lower bound: for every constant ε > 0,","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"12 1","pages":"6:1-6:45"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87522728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved Learning from Kolmogorov Complexity","authors":"H. Goldberg, Valentine Kabanets","doi":"10.4230/LIPIcs.CCC.2023.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2023.12","url":null,"abstract":"Carmosino, Impagliazzo, Kabanets, and Kolokolova (CCC, 2016) showed that the existence of natural properties in the sense of Razborov and Rudich (JCSS, 1997) implies PAC learning algorithms in the sense of Valiant (Comm. ACM, 1984), for boolean functions in P / poly , under the uniform distribution and with membership queries. It is still an open problem to get from natural properties learning algorithms that do not rely on membership queries but rather use randomly drawn labeled examples. Natural properties may be understood as an average-case version of MCSP, the problem of deciding the minimum size of a circuit computing a given truth-table. Problems related to MCSP include those concerning time-bounded Kolmogorov complexity. MKTP, for example, asks for the KT-complexity of a given string. KT-complexity is a relaxation of circuit size, as it does away with the requirement that a short description of a string be interpreted as a boolean circuit. In this work, under assumptions of MKTP and the related problem MK t P being easy on average, we get learning algorithms for boolean functions in P / poly that work over any distribution D samplable by a family of polynomial-size circuits (given explicitly in the case of MKTP ), only use randomly drawn labeled examples from D , and are agnostic (do not require the target function to belong to the hypothesis class). Our results build upon the recent work of Hirahara and Nanashima (FOCS, 2021) who showed similar learning consequences but under a stronger assumption that NP is easy on average.","PeriodicalId":11639,"journal":{"name":"Electron. Colloquium Comput. Complex.","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74434759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}